[...]... present in real data T M¨ ller et al (eds.), Mathematical Foundations of Scientific Visualization, Computer o Graphics, and Massive Data Exploration, Mathematics and Visualization, DOI: 10.1007/978-3-540-49926-8, c 2009 Springer-Verlag Berlin Heidelberg 19 20 V Pascucci et al (a) (b) (c) (d) Fig 1 (a) Orrery reproducing the hierarchical relationship between the orbits of the sun, the planets and their moons... critical points of a real functions of n independent variables Transactions of the American Mathematical Society, 27:345–396, July 1925 17 S P Morse A mathematical model of the analysis on contour-line data Journal of the Association for Computing Machinery, 15(2):205–220, 1968 18 V Pascucci and K Cole-McLaughlin Efficient computation of the topology of level sets In M Gross, K I Joy, and R J Moorhead,... of topological similarity for databases of geometric models [12] The Toporrery: Computation and Presentation of Multiresolution Topology 21 Topological information has been used to guide the construction of transfer function for volume rendering of scientific data [21, 26] A more extensive discussion of the use of the Reeb graph and its variations in geometric modeling and visualization can be found... are demonstrated on datasets of different nature, such as terrain, surface models and volumetric scientific data 2 Multiresolution Contour Trees Contour Tree of a Morse Function Let D be a triangulated domain and f : D → R be a function obtained by linear interpolation of the value of f at the vertices of D Morse theory provides a formal framework for understanding the topology of D by analyzing the... IEEE Transactions on Visualization and Computer Graphics, 10(4):385–396, 2004 5 H Carr, J Snoeyink, and U Axen Computing contour trees in all dimensions Computational Geometry: Theory and Applications, 24(3):75–94, 2003 6 A Cayley On contour and slope lines The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, XVIII:264–268, 1859 7 W de Leeuw and R van Liere Collapsing flow topology... the union of a and all integral lines that end at a The collection of descending manifolds is a complex in the sense that the boundary of a cell is the union of lower-dimensional cells Symmetrically, we define the ascending manifold A(a) of a as the union of a and all integral lines that start at a If no integral line starts and ends at a saddle, see [9], we can overlay these two complexes and obtain... School of Computing, University of Utah, UT, USA 2 Dept of Mathematics, UCLA, CA, USA, School of Engineering, University of Roma Tre, Roma, Italy Summary The Contour Tree of a scalar field is the graph obtained by contracting all the connected components of the level sets of the field into points This is a powerful abstraction for representing the structure of the field with explicit description of the... in Morse theory is the characterization of each point of D as being either regular or critical We assume that D is a simplicial complex Therefore, every k-cell c of D is the convex hull of k + 1 vertices of D Moreover, a cell c is a called face of c if its The Toporrery: Computation and Presentation of Multiresolution Topology 23 vertices are a subset of those of c If c ∈ D then all its faces must be... Topological persistence and simplification Discrete and Computational Geometry, 28:511–533, 2002 11 L De Floriani, E Puppo, and P Magillo A formal approach to multiresolution modeling In W Straßer, R Klein, and R Rau, editors, Theory and Practice of Geometric Modeling Springer, Berlin, 1996 12 J L Helman and L Hesselink Visualizing vector field topology in fluid flows IEEE Computer Graphics and Applications,... which form the cancellation forest Figure 6 shows several cancellations and the resulting trees Figure 17a shows the cancellation trees of a typical terrain data set Notice, that the cancellation trees provide a very intuitive description of the orientation and general shape of the dominate ridges and valleys in the data Even though the data structure used for cancellation trees is simple, it is also very . Russell Editors Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration With 183 Figures, 134 in Color and 15 Tables 123 Torsten Möller School of Computing Science Simon. ¨oller et al. (eds.), Mathematical Foundations of Scientific Visualization, Computer 1 Graphics, and Massive Data Exploration, Mathematics and Visualization, DOI: 10.1007/978-3-540-49926-8, c  2009. for Data Analysis and Visualization, Department of Computer Science, University of California, Davis hamann@cs.ucdavis.edu Summary. We present a highly adaptive hierarchical representation of